Material softening and strain localization in spatial geometrically exact beam finite element method with embedded discontinuity. (1st April 2017) - Pirmanšek, K.; Češarek, P.; Zupan, D.; Saje, M.
Highlights: The paper addresses strain localization in spatial geometrically exact beams due to softening of material. The formulation is classified as the embedded strong strain discontinuity approach. It is an extension of the strain-based formulation of spatial beams by Zupan and Saje[31] to include softening material. Strain fields are assumed to have singular strain peaks at the localized section. Displacements and rotations are step-like discontinuous. The continuous parts of strains are only unknown functions that need be interpolated in the formulation. Abstract: When some critical condition is reached at a material point of a solid body, a localized strain starts developing which makes the strain field discontinuous and highly accelerates local damaging of material. The present paper addresses this kind of strain localization in spatial geometrically exact beams. Here we propose a new beam finite element formulation which accounts for softening of material by applying the embedded strong strain discontinuity technology. The formulation is essentially an extension of the original strain-based formulation, upgraded such to allow for detecting the onset of strain localization and to introduce additional equations for evaluating singular strain peaks and jumps of displacements and rotations at the localized section in further deformation. The consistency condition that the equilibrium and the constitutive stress-resultants are equal is shown to be naturally suited for the implementation into the discontinuous formulation. The condition for the onset of strain localization at a beam cross-section is here related to the loss of uniqueness of the beam cross-sectional constitutive equations. If the condition for a unique inverse is violated, two solutions are possible for cross-sectional strains. In a subsequent deformation, one of the two solutions follows the softening regime of material. The discontinuous increments in strains, displacements and rotations at the softening cross-section are obtained from the equations of the structure supplemented by the consistency conditions of the softening cross-section.